To begin with we clarify the notion of weak solutions (see for instance. [4]). Definition 1. We say r a weak solution of. (1.1)(1.2)(1.3) if the following conditions are ...
No. 1]
Proc. Japan Acad., 84, Ser. A (2008)
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Existence of solutions for the nonlinear partial differential equation arising in the optimal investment problem By Ryo A BEÞ and Naoyuki I SHIMURAÞ (Communicated by Heisuke HIRONAKA,
M.J.A.,
Dec. 12, 2007)
Abstract: We are concerned with the solvablity of certain nonlinear partial differential equation (PDE), which is derived from the optimal investment problem under the random risk process. The equation describes the evolution of the Arrow-Pratt coefficient of absolute risk aversion with respect to the optimal value function. Employing the fixed point approach combined with the convergence argument we show the existence of solutions. Key words:
Absolute risk aversion; nonlinear partial differential equations; solvability.
1. Introduction. We deal with the existence of solutions for the singular parabolic partial differential equation (PDE) of the form @r @ 1 @r 2 ¼ 1þ 2 r ; ð1:1Þ @t @x r @x r ¼ rðx; tÞ in ðx; tÞ 2 T :¼ Rþ ð0; T Þ where T > 0 and Rþ ¼ fx > 0g. The unknown function r is related to the Arrow-Pratt coefficient of absolute risk aversion [11] for the optimal value function; it is natural to assume that r is positive and non-increasing. We thus impose the next condition for r. ð1:2Þ
@r ð0; tÞ ¼ 0; @x rðx; tÞ ! as x ! 1;
r 0;
where denotes non-negative constant. The derivation of (1.1) and other properties are recalled in §2. The problem (1.1)(1.2) is supplemented by the initial condition. ð1:3Þ
rðx; 0Þ ¼ r0 ðxÞ
on x 2 Rþ ;
where the non-increasing initial datum r0 belongs to H 1 ðRþ Þ and satisfies the compatibility condition (1.2). The aim of the current article is to solve 2000 Mathematics Subject Classification. Primary 35K60, 91B28; 91B30. Þ Simplex Technology Inc., Nihonbashi, Tokyo 103-0027, Japan. Þ Department of Mathematics, Graduate School of Economics, Hitotsubashi University, Kunitachi, Tokyo 186-8601, Japan.
#2008 The Japan Academy
(1.1)(1.2)(1.3) in a weak sense. To begin with we clarify the notion of weak solutions (see for instance [4]). Definition 1. We say r a weak solution of (1.1)(1.2)(1.3) if the following conditions are fulfilled. (1) r 2 L1 ð0; T ; L2 ðRþ ÞÞ \ L2 ð0; T ; H 1 ðRþ ÞÞ, @r=@t 2 L2 ð0; T ; H 1 ðRþ ÞÞ. (2) There hold for each ’ 2 H 1 ðRþ Þ and almost every 0 t T Z @r ð1:4Þ ðx; tÞ’ðxÞdx þ @t R Z 1 @r @’ @r ¼ þ 2r ’ dx: 1þ 2 r @x @x @x Rþ (3) rðx; 0Þ ¼ r0 ðxÞ in L2 ðRþ Þ. The main result of this paper is then stated as follows: Theorem 2. For any positive and non-increasing r0 2 H 1 ðRþ Þ satisfying ð@r0 =@xÞð0Þ ¼ 0 and r0 ðxÞ ! as x ! 1 with > 0, there corresponds T ¼ T ðr0 Þ > 0 such that there exists a positive solution r for (1.1)(1.2)(1.3), which is nonincreasing in x, in the sense of Definition 1. The proof of Theorem, which is performed in §3, is based on an approximation argument with the combination of fixed point approach. §4 is devoted to the classification of steady state solutions of (1.1). We conclude by Discussions in §5 with a comment on the interpretation in economics. 2. Model equation. Here we briefly sketch the derivation and survey the background issues of (1.1). The basic model we follow is due to Browne [2].
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R. ABE and N. ISHIMURA
It is assumed that there is only one risky stock available for investment, whose price Pt at time t is governed by the stochastic differential equation of Black-Scholes-Merton type [3,10]
ð2:3Þ
ð1Þ
dPt ¼ Pt ðdt þ dWt Þ; ð1Þ
where and are constants and fWt gt0 is a standard Brownian motion. There is also a risk process, which is denoted by Yt and assumed to be modeled as ð2Þ
dYt ¼ dt þ dWt ; ð2Þ
where and ( > 0) are constants and fWt gt0 is another standard Brownian motion. It is allowed these two Brownian motions to be correlated with the correlation coefficient . We prescribe 0 2 < 1 in the sequel. The company invests in the risky stock under an investment policy f, where f ¼ fft g0tT is a suitable, admissible adapted control process. T stands for the maturity date. Let Xtf denote the wealth of the company at time t with X0 ¼ x, whose evolution process is given by dXtf ¼ ft
dPt þ dYt ; Pt
X0 ¼ x:
The generator Af of this wealth process is then expressed as ðAf gÞðx; tÞ ¼
@g @g þ ðf þ Þ @t @x 1 2 2 @2g 2 þ ðf þ þ 2fÞ 2 : 2 @x
Suppose that the investor wants to maximize the utility uðxÞ from his terminal wealth. The utility function uðxÞ is customarily assumed to satisfy u0 > 0 and u00 < 0. Let V ðx; tÞ ¼ supf E½uðXTf Þ j Xtf ¼ x. Then the Hamilton-JacobiBellman equation becomes ð2:1Þ
supfAf V ðx; tÞg ¼ 0;
V ðx; T Þ ¼ uðxÞ:
f
Suppose that (2.1) has a classical solution V with @V =@x > 0, @ 2 V =@x2 < 0. We then infer that ð2:2Þ
ft
@V =@x ; ¼ 2 2 @ V =@x2
where f ft g0tT denotes the optimal policy. Placing (2.2) back into (2.1) we obtain
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@V @V 1 2 ð@V =@xÞ2 þ @t @x 2 @ 2 V =@x2 1 @2V þ 2 ð1 2 Þ 2 ¼ 0 for 0 < t < T 2 @x V ðT ; xÞ ¼ uðxÞ:
Browne [2] shows that (2.3) possesses a solution in the case uðxÞ ¼ ð=Þex with positive constants , , . This utility has constant absolute risk aversion parameter ; precisely stated, u00 ðxÞ= u0 ðxÞ ¼ . Abe [1] made a preliminary research whether (2.3) has other solutions. Here we proceed further in the analysis of (2.3). Let vðx; tÞ be defined by V ðx; T tÞ ¼ vðEðx þ F tÞ; GtÞ, where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 2 E¼ ; F ¼ ; G¼ : 2 ð1 2 Þ 2 2 It follows that after a calculation ð2:4Þ
@v @ 2 v ð@v=@xÞ2 ¼ 2 2 ; @t @x @ v=@x2
vðx; 0Þ ¼ uðE 1 xÞ:
The study of this singular parabolic PDE (2.4) seems interesting. Here we additionally introduce @v @ 2 v=@x2 @ ð2:5Þ rðx; tÞ :¼ log ðx; tÞ: ¼ @x @x @v=@x A little tedious computation then finally leads us to the equation (1.1). It should be noted that (2.5) is related to the coefficient of absolute risk aversion. 3. Proof of Theorem. Now we prove Theorem 2. Since we seek for a solution r which tends to ð> 0Þ as x ! 1, we make a translation r 7! r þ so that we consider the next problem. ( ! ) @r @ 1 @r 2 ¼ 1þ ðr þ Þ ; ð3:1Þ @t @x ðr þ Þ2 @x r ¼ rðx; tÞ in ðx; tÞ 2 T @r @r ð0; tÞ ¼ 0; ðx; tÞ 0 for ðx; tÞ 2 T @x @x r!0 as x ! 1 for 0 < t < T rðx; 0Þ ¼ r0 ðxÞ for x 2 Rþ : Fix L > 1 and let LT :¼ ð0; LÞ ð0; T Þ. We approximate r0 2 H 1 ðRþ Þ by rL0 2 C 1 ½0; L with @rL0 =@xð0Þ ¼ 0 and rL0 ðLÞ ¼ 0. We introduce the convex set E L defined as E L :¼ fr 2 C 1 ðLT Þ j 0 r K; K @r=@x 0 in LT ;
No. 1]
Optimal investment problem
@rð0; tÞ=@x ¼ 0; rðL; tÞ ¼ 0 for 0 < t < T ; rðx; 0Þ ¼ rL0 ðxÞ for 0 x Lg; where K :¼ 2ðkr0 kC 1 ½0;L þ 1Þ. Take s 2 E L and try to find a solution r 2 E L of the problem ! 2 @r 1 @2r 2 @r ¼ 1þ ð3:2Þ 2 3 2 @t ðs þ Þ @x ðs þ Þ @x @r in ðx; tÞ 2 LT : @x A standard theory of PDE tells us that an a-priori bound of r and @r=@x implies the existence of solutions [5]. We first see that a simple application of the maximum principle yields 0 r krL0 kL1 ð0;LÞ . Moreover we derive that the equation for q :¼ @r=@x becomes ! @q 1 @2q 2 @s @q ¼ 1þ þ 2q 2 3 2 @t @x @x @x ðs þ Þ ðs þ Þ ! @q 3 @s 2 þ 1 q2 2ðr þ Þ @x ðs þ Þ3 @x 2ðr þ Þ
in ðx; tÞ 2 LT : Since N :¼ 33 K þ 1 3ðs þ Þ3 ð@s=@xÞ þ 1 0, it follows that K @r=@x 0 in LT with T :¼ ð4NKÞ1 by virture that 2NK 2 t 21 K gives a lower barrier. Gradient bound at x ¼ L is provided by a supersolution 21 Kðx L þ 1Þ2 þ 21 K for (3.2) on L 1 < x < L. Thus we deduce that krð; tÞkC 1 ½0;L K for every 0 t T , which ensures the existence of a solution to (3.2). We define an operator B : E L ! E L by s 7! r ¼ Bs is a solution of (3.2). Parabolic regularity shows that B is compact. Since E L is a closed convex set in a Banach space C 1 ðLT Þ we infer that there exists a fixed point rL of B in E L thanks to the Leray-Schauder fixed point theorem. See Corollary 11.2 of [6]. Now that we have found a non-increasing solution rL for the equation (1.1) on LT with rL ðLÞ ¼ , we want to take a limit of L ! 1. To do so we first extend rL to be defined on Rþ by setting rL ðx; tÞ ¼ for x L. Take any monotone increasing sequence 0 < L1 < L2 < < Ln < ! 1 and we write rn ðx; tÞ ¼ rLn ðx; tÞ for simplicity. Plugging ’ ¼ rn rm 2 H 1 ðRþ Þ ðn > mÞ into (1.4) for the equations of rn and rm respectively, and subtracting term by term, we compute
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2 @ 1 d 2 kðrn rm ÞðtÞkL2 ðRþ Þ þ ðrn rm ÞðtÞ 2 þ 2 dt @x L ðR Þ Z @rn =@x @rm =@x ¼ r2n þ r2m 2 2 þ r r R n m @ðrn rm Þ dx @x 1 @rm 2 þ 1þ 2 þ rm ðrn rm Þ x¼Lm rm @x K 2ðK þ Þ 4 þ 1 kðrn rm ÞðtÞkL2 ðRþ Þ @ ðrn rm ÞðtÞ 2 þ @x L ðR Þ 1 þ 1 þ 2 K þ 2 jðrn ÞðLm ; tÞj 2 1 @ ðrn rm ÞðtÞ 2 þ 2 @x L ðR Þ 2 K þ 2ðK þ Þ2 4 þ 1 kðrn rm ÞðtÞk2L2 ðRþ Þ 1 2 þ 1 þ 2 K þ jðrn ÞðLm ; tÞj: Gronwall lemma implies immediately that kðrn rm ÞðtÞk2L2 ðRþ Þ 2 Z t @ ðr þ eCðt Þ r Þð Þ m @x n 2 þ d 0 L ðR Þ eCt kðr0 Þn ðr0 Þm k2L2 ðRþ Þ Z t 1 2 þ 1þ 2 Kþ eCðt Þ jðrn ÞðLm ; Þjd ; 0 where C :¼ 4ðK þ Þ2 ð4 K þ 1Þ2 . Since kðr0 Þn ðr0 Þm k2L2 ðRþ Þ ! 0 as well as jðrn ÞðLm ; Þj ! 0 as n; m ! 1, we see that frn g is a Cauchy sequence in L1 ð0; T ; L2 ðRþ ÞÞ \ L2 ð0; T ; H 1 ðRþ ÞÞ. The limiting function r is seen to verify (1.4) for each ’ 2 H 1 ðRþ Þ and almost every 0 t T . The existence of solution r claimed in Theorem 2 is thereby established. 4. Steady state solutions. In this section we analyze the structure of steady state solutions of (1.1); that is, we determine the set of solutions for 1 @r 1þ 2 r2 ¼ C; r ¼ rðxÞ; r @x where C denotes a constant independent of x and t. There are three possibilities according to the sign of C. We note that, however, the first two cases are
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R. ABE and N. ISHIMURA
meaningless for the economics because r takes a negative value. If C > 0 then we write C ¼ M 2 to obtain 1=M 2 1 1 rðxÞ ð4:1Þ 3 tan1 þ M M M rðxÞ 2 1=M 1 1 rð0Þ : þ ¼x tan1 M M3 M rð0Þ It follows that rðxÞ M 2 x1 as x ! 1. If C ¼ 0 then we have ð4:2Þ
1 1 1 1 ¼x : 3 rðxÞ 3rðxÞ rð0Þ 3rð0Þ3
It follows that rðxÞ ð3xÞ1=3 as x ! 1. Consequently if C 0 then there is no steady state solution suitable to the finance. Only the next last case fits into our requirement. If C < 0 then we write C ¼ M 2 to obtain rðxÞ M 1=M 2 1 þ M 2 ð4:3Þ log þ rðxÞ þ M rðxÞ 2M 3 rð0Þ M 1=M 2 1 þ M 2 ¼xþ log þ ; 3 rð0Þ þ M rð0Þ 2M provided rðxÞ 6¼ M. In this case rðxÞ M 2 x1 as x ! 1. It is also clear that rðxÞ M gives one of steady state solutions, which has a character of constant absolute risk aversion. It should be noted that the last steady state solutions correspond to those presented in [2]. 5. Discussions. The solvability is discussed for certain singular parabolic partial differential equation (PDE), which is related to the ArrowPratt coefficient of absolute risk aversion for the optimal value function. We prove the existence of solutions, which tend to a positive constant . However, the analysis of steady state solutions exhibited in §4 indicates that the case ¼ 0 may be possible. The problem whether the equation admit a solution which tends to zero or not is worth further investigation. We will return to this topic from the computational standpoint in the next paper [8]. The singular parabolic PDE of the form (2.3) or (2.4) has been often observed in the stochastic control theory. Indeed as mentioned in Hipp [7] the achievement of Browne [2] is one of first papers on this basis appeared in insurance mathematics, which is now an important area of applications of the stochastic control. We therefore believe that the advanced qualitative study of such singular PDEs is indispensable from the viewpoint of applications.
[Vol. 84(A),
For more details on stochastic control applied in insurance mathematics, see for instance a nice review of Hipp [7] and the references cited therein. We should point out, however, that the analysis of these singular equations is much more challenging than that of usual possible nonlinear Black-Scholes equations (see for example [9,12]). We hope that our paper has made a first step toward the better understanding of these PDEs. Acknowledgments. Thanks are due to the referee for careful reading and valuable suggestions, which greatly helps in improving the manuscript. The work of the second author (NI) is partially supported by the Inamori Foundation through the Inamori Grants for the fiscal year 2006–2007. References [ 1 ] R. Abe, Dynamic optimal investment problem in ALM via the theory of partial differential equations, Thesis for the Master-course degree, Graduate School of Economics, Hitotsubashi University, (2006). (in Japanese). [ 2 ] S. Browne, Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin, Math. Operations Research 20 (1995), no. 4, 937–958. [ 3 ] F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Economy 81 (1973), 637–659. [ 4 ] L. C. Evans, Partial differential equations, Amer. Math. Soc., Providence, RI, 1998. [ 5 ] A. Friedman, Partial differential equations of parabolic type, Krieger Publishing Company, Florida, 1983. [ 6 ] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer Classics in Math., Springer-Verlag, 2001. [ 7 ] C. Hipp, Stochastic control with application in insurance, in Stochastic methods in finance, 127–164, Lecture Notes in Math., 1856, Springer, Berlin. [ 8 ] H. Imai, N. Ishimura and M. Kushida, Numerical treatment of a singular nonlinear partial differential equation arising in the optimal investment. (Preprint). [ 9 ] H. Imai, N. Ishimura, I. Mottate and M. A. Nakamura, On the Hoggard-Whalley-Wilmott equation for the pricing of options with transaction costs, Asia-Pacific Financial Markets 13 (2007), 315–326. [ 10 ] R. C. Merton, Theory of rational option pricing, Bell J. Econ. Manag. Sci. 4 (1973), 141–183. [ 11 ] J. W. Pratt, Risk aversion in the small and in the large, Econometrica 32 (1964), 122–136. [ 12 ] P. Wilmott, S. Howison and J. Dewynne, The mathematics of financial derivatives, Cambridge Univ. Press, Cambridge, 1995.
